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Hello, welcome to another module in this massive
open online course on probability and random
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variables for wireless communications.
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So we are looking at independence, the concept
of independence and independent events and
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let us look at one final example to understand
it.
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We have already looked at several examples
in this context to better understand the concept
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of independence and more importantly, its
relevance in the context of wireless communication.
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Remember, in the beginning of this course
itself and in the introductory module also,
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we have said that this is not simply an abstract
course on probability and random variables.
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This is not a course, this is simply the mathematics
of probability and random variables.
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So one of the important aspects of this course
is to explore and examine the impact or examine
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the relevance of probability and random variables,
random processes in the context of communication,
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especially digital communication systems and
wireless, modern wireless communication systems.
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So that it bridges or basically builds a bridge
between these abstract concepts in probability
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and random variables and the practical applications
of these concepts in the context of modern
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communication systems, both high-speed digital
communication systems as well as high-speed
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wireless communication systems, 3G and 4G
wireless communication systems.
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So in this module, let us briefly at a very
high-level look at another, very important
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impact or relevance of this concept of independence
for independent events in a wireless communication
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system.
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And to draw this example from the context
of 3G, 4G wireless communication system, some
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of you who have seen previous courses of wireless
communication systems, might have already
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seen this.
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This is a concept in the context of what is
known as diversity.
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So let us look at an example of independence
in the context of wireless communication systems.
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So I am looking at
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independence in the context of wireless systems.
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And especially in the context of 3G and 4G
wireless communication systems.
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And what we would like to look at?
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We
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would like to look at what is known as the
probability of a deep fade event?
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Some of you might already be familiar with
this concept of a deep fade.
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But if you are not familiar, let me explain
this to you briefly at a high level.
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The wireless communication signal that is
received at the receiver in a wireless communication
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system is the result of superposition of multiple
signal copies that arise for instance from
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a direct signal that is coming from the base
station or also multiple copies that are coming
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from the reflections from various buildings,
various trees and other objects in the wireless
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government.
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So when these signals superpose at the receiver,
there is interference.
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And this interference can be either constructive
in nature or it can be destructive which causes
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fading.
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So the interference in the wireless communication
environment, that results in fading.
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That is, which is basically either an amplification
of the received signal power or the dip in
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the received signal power.
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This process is known as fading.
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And when the received signal power dips significantly
below a certain threshold, that is below a
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certain threshold, below the noise threshold
at the receiver so that communication is not
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possible, that is known as a deep fade event.
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So a deep fade event, to put it simply that
is a significant drop in the received signal
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power in a fading wireless channel.
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So the significant dip in the received power
in a fading wireless channel or in a fading
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wireless system, this is known as a deep fade
event.
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Obviously, we would like to avoid a deep fade
event because whenever the signal is in a
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deep fade, then communication is not possible
between the transmitter and the receiver because
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simply the received signal power is very low.
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And one way to avoid a deep fade is through
what is known as multiple antennas.
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By increasing the number of antennas at the
transmitter and receiver, one can significantly
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reduce the probability of deep fade.
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Let us look at how this happens.
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Let us say, I have a transmitter which is
let us say something like my base station.
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So let us say I have my transmitter, I have
my receiver.
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Let us say my transmitter has a single antenna
but my receiver, let us say has multiple antennas.
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So my receiver has antennas 1, 2, up to L
antennas.
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So these are multiple antennas at the receiver.
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These are L antennas.
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I have a single antenna at the transmitter.
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Although in the most general case, one can
have multiple antennas both at the transmitter
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and the receiver.
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For the purpose of this simple example, we
are looking at a single antenna at the transmitter
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and multiple antennas, capital L antennas
at the receiver.
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This is known as a SIMO system.
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This is a single input multiple output system
because I have a single antenna at the transmitter,
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that is the input, multiple antennas at the
receiver, that is the output.
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This is known as a SIMO system or a single
input multiple output system.
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So this is known as a SIMO system where SIMO
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stands for single input multiple output.
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And therefore, in this system, we have capital
L links.
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We have 1, 2, up to L links.
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So we have link 1, 2, up to L links.
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So what do you mean by L links?
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We have a link between the transmit antenna
and each receiver antenna.
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So in this system what we are saying is we
have basically L different links.
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Because I have a single transmit antenna and
L receiver antennas, one can consider the
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link between, one can consider the channel
between the single transmit antenna and each
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of the receiver antennas as a link.
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So we have a single transmit antenna, L receiver
antennas.
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So there are L links in this system.
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And now therefore, the probability that each,
let Ei, let this denote the event that link
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i is in deep fade.
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What we are saying is let Ei denote the event,
let this quantity Ei denote that the link
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i is in deep fade.
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So each of these links can be in deep fade
. That is the SNR received over this link
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is below a certain threshold or below the
noise threshold and we can say that then the
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link i is in deep fade and let this Ei denote
the event that link i is in deep fade.
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Now therefore one can ask the question, when
is this wireless communication system, when
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is the whole system in a deep fade?
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The whole system is in a deep fade that is
no communication is possible naturally if
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all the links are in a deep fade.
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In this system with L links, the communication
between the transmitter and receiver is not
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possible if all these links that is each of
these L links is in a deep fade.
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So the system, the entire wireless
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system is in
a deep fade
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if all the links , this is the system, a wireless
communication system, multiple antenna wireless
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communication system is in a deep fade if
all the links are in a deep fade.
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Now the probability that each of these individual
links is in a deep fade is equal to 1 over
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SNR.
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This is the result from wireless communication
which I am not deriving here.
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The probability that each of these, the probability
that link Ei, the probability of this event
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is Ei that link i is in deep fade is 1 over
SNR which is basically 1 over P over Sigma
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Square where P is the transmitted power and
Sigma Square is the noise power at the receiver.
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So this P, let me also clarify, this P equals
the transmitted power and Sigma Square equals
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noise power.
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P over Sigma Square is the SNR and the probability
that link Ei, the probability of Ei that is
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the event that link i is in deep fade is 1
over SNR and we are asking the question that
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what is the probability that the entire system
is in deep fade?
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That is, probability of deep fade of the system
is basically we said the probability that
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all the links are in deep fade which is the
probability that E1 intersection E2 intersection
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E3 so on and so forth up till EL.
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That is the probability of the joint event.
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Therefore the probability that the system
is in deep fade is the probability that all
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the links are in deep fade.
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Now I am going to use the independent assumption
to write this as the probability of E1 times
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the probability of E2 times the probability
of EL and this is the key.
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This is using
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assuming all the links are fading.
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So what we are saying is the probability that
the system is in fade is the joint event that
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is E1 intersection E2 and so on intersection
EL.
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That is the joint event that all the links
are in a deep fade and the probability of
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this joint event we are saying is the product
of the individual deep fade events.
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That is, the probability of E1 times the probability
of E2 times so on the probability of EL given
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that all the links are fading independently.
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And previously, we had seen that the probability
that each of the links, each of the individual
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links is in a deep fade is 1 over SNR.
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Therefore the probability of this joint event
is basically simply given as 1 over SNR times
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1 over SNR times times 1 over SNR which is
equal to 1 over SNR raised to the power of
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L.
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Therefore the probability of deep shade, if
I call this as PDF this is equal to 1 over
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SNR raised to the power of capital L. And
this is an important property in wireless
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communication systems.
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Especially in modern 3G, 4G wireless communication
systems, this EDF, this is the probability
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of deep fade.
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This is
the probability of deep fade which is basically
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1 over SNR raised to the power of capital
L. So we are saying that as the number of
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antennas at the receiver increases, the probability
of the fade that is the probability that the
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system is in a deep fade or the signal power
at the receiver is below the noise threshold
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decreases as 1 over SNR raised to the power
of capital L.
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And therefore, having a large number of antennas
helps improve the efficiency of communication
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or helps decrease the error rate, helps decrease
the bit error rate of communication because
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the probability of deep fade decreases as
we have seen in this.
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We are employing the assumption of independently
fading channel.
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This property is known as diversity.
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The probability where the bit, the probability
of is decreasing as 1 over SNR raised to the
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power of L, this is known as diversity.
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And this leads to, diversity leads to a significant
decrease in what is the bit error rate or
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the BER of wireless communication.
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So what we are saying is, having multiple
antennas, the probability that the system
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is in a deep fade decreases as 1 over SNR
raised to the power of L.
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So the larger the number of receiver antennas,
the probability of deep fade becomes correspondingly
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lower and therefore this leads to a significant
decrease in the probability of deep fade which
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in turn leads to a significant decrease in
the bit error rate of wireless communication.
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This is an important property in 3G or 4G
wireless communication systems very imply
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multiple antennas at both, the base station,
either the base station or the handset and
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if possible, both at the base station and
also the handset.
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The main reason is because having multiple
antennas leads to a significant improvement
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in reliability of wireless communication.
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This is achieved through the property of principle
of diversity and the principle of diversity
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is fundamentally based on the assumption of
these independently fading place.
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So this is a key impact or a key relevance
of the concept of independence and independent
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events in the context of wireless communication,
especially modern 3G, 4G wireless communication
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systems.
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So with this, let us stop this module here
and I hope I have explained an important aspect
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of independence in the context of wireless
communication.
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Thank you very much.